The General Term of a G.P.

Transcript

0:00Introduction

Hello everyone! Okay, a quick puzzle to start. Take five, and keep doubling it: five, ten, twenty, forty, and so on. What is the eighth number in this list? Five times two to the power eight? Hmm, are you sure? Today we learn the general term of a geometric progression, and it settles this in one line. Questions on G P come up often in JEE, so let's begin.

0:29What is a G.P.?

So, here is our list, and the three dots mean it goes on. Each term is two times the one before. In an A P, we subtracted. Here, what do we do instead? We divide, yes, exactly! So a geometric progression, or G P, is a sequence in which each term divided by the one before is the same constant, r, the common ratio. T one, a capital T with a small one below it, is the first term. The fraction bar means divided by, and the dots in the middle mean and so on. So r equals T two over T one, equals T three over T two. Okay? Now tell me. Can a term of a G P be zero? No, never! We divide by every term, and we cannot divide by zero. So every term of a G P must be non-zero. Clear?

1:21The general term

Now, let's build the terms. Small a is the first term, and r is the common ratio. T one is a, and T two is a r. T three is a r squared; the small two up high is the power. T four is a r cubed. See the pattern? The power of r is one less than the position. Right? So, tell me. In the eighth term, what is the power of r? Seven, very good! So in general, T n equals a times r to the power n minus one. This is the general term. Write it down. The classic slip is a r to the power n: one multiplication too many! Got it? Now, our puzzle. Here a is five and r is two. So T eight is five times two to the power seven, not eight; the dot means times. That is five times one hundred and twenty eight: six hundred and forty. The power eight would double the right answer! With me?

2:20Properties of a G.P.

Now, a quick test. Three numbers a, b, c, in this order; here a is just the first of the three. How do you check that they are in G P? Yes, the ratios must be equal: b over a equals c over b. Cross multiply, and b squared equals a c. The double arrow pointing both ways means if and only if. Right? Next, what keeps a G P as a G P? Multiply every term by the same non-zero constant, take reciprocals, or raise every term to the same power. And multiply two G Ps term by term: still a G P, okay? One link with the last chapter. If a G P has positive terms, the logs of its terms form an A P! Why? Take logs of b squared equals a c: two log b equals log a plus log c, the A P test. Positive terms only: the log of a negative number or zero is not defined. Clear?

3:22Terms equidistant from the ends

Now take a finite G P with n terms. T k is the k-th term from the start; the term with subscript n minus k plus one is the k-th from the end. Their product is always T one times T n, first times last. Step in from the left, gain a factor r; from the right, lose one. They balance! Okay? More generally, T p times T q equals T r times T s, whenever p plus q equals r plus s. Only the positions matter. With me? So the terms pair up from the two ends, each pair giving T one times T n. Hence, for a G P of positive terms, the product of all n terms is T one times T n, in brackets, to the power n over two. Careful: with a negative ratio and an odd number of terms, this can give the wrong sign. For one, minus two, four, the product is minus eight, not eight. Right?

4:19Topper's trick: choose the terms symmetrically

Now a trick that toppers use. For three numbers in G P with a known product, do not take a, a r, a r squared. Take a over r, a, and a r, with a in the middle. Now tell me, what is their product? It is a cubed, exactly! The r below the line cancels the r on top, so the product gives the middle term straight away. Nice, isn't it? For four terms, keep the balance: a over r cubed, a over r, a r, a r cubed. But careful! From a over r to a r, you multiply by r squared. So the common ratio here is r squared, not r. And r squared is positive, so this form only fits a G P with a positive common ratio. Easy to miss, okay?

5:07Example 1: which term of 2, 2√2, 4, ... is 128

Okay, Example one, an easy one from the course. Which term of the G P two, two root two, four, and so on, is one hundred and twenty eight? The tick-shaped sign is the square root sign. Pause the video, and try it first. Done? Okay, let's check. The first term a is two, and r is two root two over two, which is root two. Clear? Set T n equal to one hundred and twenty eight: two times root two, in brackets, to the power n minus one. The arrow means this gives: divide by two, and root two to the power n minus one is sixty four. With me? Now, sixty four is two to the power six. So n minus one is six, and n is seven. Right? No, no, careful! The base here is root two, not two. Root two squared is two, so two to the six is root two squared, all to the power six: root two to the power twelve. The big brackets just group it. Okay? So n minus one is twelve, and n is thirteen. One hundred and twenty eight is the thirteenth term. And if you wrote twelve, you forgot the minus one, like in our puzzle. That's it!

6:23Example 2: three numbers in GP, sum 21, product 216

Now Example two, a little harder, also from the course. Three numbers in G P have sum twenty one and product two hundred and sixteen. Find them. Hint: use the topper's trick. Pause the video, and try it. Done? Let's check. Take the numbers as a over r, a, a r. Their product is a cubed, so a cubed is two hundred and sixteen, which gives a equals six. Now the sum: six over r, plus six, plus six r, equals twenty one. With me? Multiply through by r: six plus six r plus six r squared equals twenty one r. Bring everything to one side, and divide by three: two r squared, minus five r, plus two, equals zero. Got it? Factorise: two r minus one, times r minus two, equals zero. So r is two, or r is one half. Two values of r! So, two different sets of numbers? No, careful! Just one set. With r equal to two: three, six, twelve. With r equal to one half: twelve, six, three. The same numbers, in reverse order. Clear? Quick check: three plus six plus twelve is twenty one, and three times six times twelve is two hundred and sixteen. Perfect!

7:48Quick revision: ar^(n-1), b^2 = ac, and a/r, a, ar

So, quick revision. The n-th term is a times r to the power n minus one: the power is one less than the position. Three numbers a, b, c are in G P if and only if b squared equals a c; and T p times T q equals T r times T s when p plus q equals r plus s. And choose three terms as a over r, a, a r, or four terms with common ratio r squared, which must be positive. Now go and try the practice problems on this page. All the best!